Matrix Jeopardy

Terminologies

100

uses determinants to solve a system of n linear equations in n variables

Cramer's Rule

200

It describes the entire solution set of a linear equation.

Parametric representation

300

A square matrix that is obtained from the identity matrix by a single elementary row operation.

elementary matrix

400

A determinant of the matrix obtained by deleting the ith row and jth column of the given matrix. 

Minor of the element a_ij

500

A matrix that is obtained by transposing the cofactors of the given matrix.

Adjoint

Gaussian, LU, Minor, Cofactor and Adjoint

100

Find the solution set of the system of linear equations represented by augmented matrix.

[ 1   -1    0    3]

[0     1   -2    1]

[0     0    1   -1]

 

SS: {2, -1, -1}

200

Find AB where

A = [ 0   -1    0]     and B = [ 2     4]

      [ 4    0    2]                  [-3     2]

       [8    -1    7]                  [ 1    -5]    

AB = [3      -2]

        [10      6]

        [26     -5]

300

Given the matrix,

[ -6  -2  -1 ]

[  5   1  -6 ] 

[ -4  -6  -6 ]

what is the cofactor of 5? 

-6

400

What is the LU factorization of the following matrix?

[ 2  3 ]

[ 4  9 ]

[ 1  0 ] [ 2  3 ]

[ 2  1 ] [ 0  3 ]

500

What is the adjoint of the following matrix?

[ 2  -1  3 ]

[ 0   5  2 ]

[ 1  -1 -2 ]

[ -8   -5   -17 ]

[ 2    -7    -4 ]

[ -5    1    10]

Inverse and Determinants

100

Compute A^-2 where 

A = [2   7]

      [-5  6]

[1/2209      -56/2209]

[40/2209    -31/2209]

200

Let A be a 3x3 matrix with det(A) = 2 and B = 3A. What is the det(B)?

det(B) = det(3A)

= (3³)*det(A) = 27*2

det (B) = 54

300

Solve for the determinant:

[ 2    -4     6 ]

[ -4    6    -8 ] 

[ 6    -8    10 ]

Is it singular or non-singular?

determinant = 0

(singular matrix)

400

Use expansion by cofactors to find the determinant of the matrix

[x      y       1]

[-2    -2     1]

[1       5     1]

-7x+3y-8

500

Find |AB| where

A = [-1    2     1]    and     B = [-1    0    0]

      [1     0     1]                     [0     2    0]

      [0     1     0]                     [0     0    3]

-12

Applications

100

Given the following system of linear equations,

8x + 7y = -10

-4x - 2y = 8

Using the Cramer's rule, what is the value of D_y?

D_y = 24

200

Determine whether these points A(5, -2), B(4, -1) and C(1, 2) are collinear.

Yes, these points are collinear.

300

Find the area of parallelogram with vertices (1,1), (-4,5), (-2,8) and (3,4)

Area = 23 square units

400

Given the matrix, find its inverse using adjoint.

[1     2    3]

[0     1   -1]

[2     2     2]

[-2/3     -1/3        5/6]

[1/3        2/3      -1/6]

[1/3       -1/3      -1/6]

500

Find the volume of tetrahedron with the vertices (0, 0, 0), (0, 2, 0), (3, 0, 0) and (1, 1, 4)

4 cubic units

Vector Spaces

100

Find the cross product of u and v where

u = 2i +9j - 3k and v = -4i + i -5k

-42i +22j +38k

200

Find the dot product of u and v where 

u = (1, 1/8, 2/5) and v = (0, 1/4, 1/5)

89/800

300

Find the angle between vectors u =(0,1,0,1) and v = (3,3,3,3)

45 degrees of pi/4

400

Find the eigenvalues and eigenvectors of 

A = [2    -12]

      [1      -5]

eigenvalues are -1 and -2

eigenvectors:

[4]   and [3]

[1]         [1]

500

Write v as a linear combination of x, y and z where

v = (x, y, z), x = (1, 0 ,-2), y = (0, 1, 1) and 

z = (-1, -1, 2)

SS = {3x-y+z, 2x+z, 2x-y+z}